Multipole-based modeling

Mechatronic one-gate

In mechatronics, multipole- based modeling is a uniform representation of technical systems with a multidisciplinary character (multi-domain systems). Based on the modeling with concentrated substitute elements, as well as general maintenance and balance laws, system models with performance-preserving interconnection laws can be created (electro-analog networks, generalized networks in mechatronics). A classic example of pure electrical systems are the kirchhoff networks . Historically, this form of modeling goes back to James Clerk Maxwell. In 1873 he developed very detailed mechanical analogies to electrical phenomena. In his impedance analogy, he linked theForce and electrical voltage as analog quantities.

General

In multipole-based modeling, concentrated network elements ( lumped elements ) are interconnected via their gate terminals. A reciprocal exchange of energy takes place between the individual network elements. Depending on the number of existing gate terminals of each network element , one speaks of a one- gate , two- gate, three-gate or multi-gate. The energy exchange of a network element may always by the two elementary network variables (door sizes), the flow size (of English. Flow ) and the difference quantity (of English. Effort ) are described. The interconnection of the individual network elements is carried out using generalized Kirchhoff's laws ( node set, mesh set ). ${\displaystyle f(t)}$ ${\displaystyle e(t)}$

Formation laws for the system variables

When forming all the necessary system variables, it is initially assumed that an energy change in n-dimensional space can always be expressed by a Massieu-Gibbs function (according to Josiah Willard Gibbs ). Mathematically, we speak of a Pfaff form or linear differential form

${\displaystyle dE=\sum _{j=1}^{n}{i^{j}\ dq^{j}}}$.

Within a physical domain there are normally exactly two summands of the Massieu-Gibbs function, which are generally described by their incomplete differentials.

${\displaystyle \delta E=\delta E_{P}+\delta E_{T}=i_{T}\ \delta q_{P}+i_{P}\ \delta q_{T}}$

The two intensity quantities and form the required flow and difference quantities ( power-conjugated variables ). ${\displaystyle i_{T}}$ ${\displaystyle i_{P}}$

${\displaystyle e(t):=i_{T}}$

${\displaystyle f(t):=i_{P}}$

The product of the two intensity quantities always results in an output in the respective physical domain.

${\displaystyle P:=e(t)\cdot f(t)}$

Every single independent change in energy is expressed by a pair of energy conjugate variables . ${\displaystyle \delta E}$

${\displaystyle \delta E=i\ \delta q}$

However, the concept of quantity is not yet sufficient to uniquely characterize all forms of energy. For forms of energy in which field-like quantities are involved, there is no simple set-like relationship. For this purpose, the concept of quantity size is expanded to include the concept of extensive size with the following rules .

Education rules

• There is an extensive variable for every form of energy.
• Any quantity size is also extensive.
• Not every extensive quantity is a quantity quantity.

The following properties are thus assigned to the two set-type system variables.

variable	property	Surname	Formula symbol
${\displaystyle q_{P},q_{T}}$	are extensive variables
${\displaystyle q_{P}}$	is a quantity quantity	Primary size	${\displaystyle X}$
${\displaystyle q_{T}}$	is not a quantity size	Extensum	${\displaystyle Ex}$

Quantity and intensity sizes

Exactly four system variables can always be formed within a physical domain, two quantity quantities and two intensity quantities. Based on the primary variable, the three remaining system variables can be clearly derived mathematically.

Their metrological properties are characterized by their respective index.

P - for through (from Latin per , English through )

T - for over (from Latin trans , English across )

step	System variable	Formula symbol	equation	property	Energy variable	Network variable
0	Primary size	${\displaystyle X}$	-	P quantity	${\displaystyle q_{P}}$	-
1	Difference size	${\displaystyle Y}$	${\displaystyle Y={\frac {\delta E_{P}}{\delta X}}}$	T intensity	${\displaystyle i_{T}}$	${\displaystyle e(t)}$
2	Flow size	${\displaystyle I_{M}}$	${\displaystyle I_{M}={\frac {d}{dt}}X}$	P intensity	${\displaystyle i_{P}}$	${\displaystyle f(t)}$
3	Extensum	${\displaystyle Ex}$	${\displaystyle Ex=\int Ydt}$	T quantity	${\displaystyle q_{T}}$	-

Constitutive laws

The constitutive laws link the four system variables reciprocally with one another. This results in two energy storage elements and two dissipative elements. If one proceeds from the model concept of the concentrated substitute elements, then the four constitutive equations can each be assigned to a mechatronic one- port . The following relationship results for linear component relationships:

Surname	equation	Bauelent	property
capacitive law	${\displaystyle C:={\frac {q_{P}}{i_{T}}}}$	mechatronic capacity	Energy storage
inductive law	${\displaystyle L:={\frac {q_{T}}{i_{P}}}}$	mechatronic inductance	Energy storage
resistive law	${\displaystyle R:={\frac {i_{T}}{i_{P}}}}$	mechatronic resistance	Energy converter
memristive law	${\displaystyle M:={\frac {q_{T}}{q_{P}}}}$	mechatronic memristor	Energy converter

Overview display

The laws of formation for the system variables as well as the constituent laws for the network components can be shown together very clearly in a simple overview scheme.

history

The earliest mechanical-electrical analogy goes back to James Clerk Maxwell . In his considerations, he first linked mechanical force and electrical voltage as analog quantities (FU analogy), but without using the term impedance. This was only coined by Oliver Heaviside in 1886 , long after Maxwell's death. The idea of ​​complex impedance was then introduced by Arthur E. Kennelly in 1893. From 1900 the mechanical-electrical analogy belonged to the standard analysis methods. With the development of analog computing technology from 1923 onwards, Vannevar Bush gave analogy relationships a new boost. In 1932 Walter Hähnle published a comprehensive contribution to the representation of electromechanical structures through purely electrical circuit diagrams, almost at the same time as Floyd A. Firestone (1933). The concept of these two publications was based largely on the FI analogy (analogy between force and current), i.e. a reversal of the analogy coined by Maxwell. A modification of this FI analogy was proposed in 1955 by Horace M. Trent. Ultimately, this mathematical representation by Henry M. Paynter in 1960 led to the bond graphs, which are still used very successfully today . Under Barkhausen , the Dresden School of Acoustics successfully established itself with Reichardt , Kraak and Lenk . Arno Lenk first published a textbook here in 1971. about electromechanical systems. A similarly comprehensive approach can be found in 1979 by Peter E. Wellstead . In addition to multipole-based modeling, Wellstead also uses the Lagrange and Hamilton formalism for the first time.

literature

• Rüdiger G.Ballas, Günther Pfeifer and Roland Werthschützky: Electromechanical systems in microtechnology and mechatronics: Dynamic design - basics and applications . 2nd Edition. Springer, 2009, ISBN 978-3-540-89317-2 .
• Gottfried Falk, Wolfgang Ruppel: Energy and Entropy . Springer, Berlin / Heidelberg / New York 1976, ISBN 3-540-07814-2 .
• Gottfried Falk: Physics: Number and Reality: The conceptual and mathematical foundations of a universal quantitative description of nature. Birkhäuser Verlag, Basel 1990, ISBN 3-7643-2550-X .
• Jörg Grabow: Generalized networks in mechatronics . Oldenbourg Wissenschaftsverlag, 2013, ISBN 978-3-486-71261-2 .
• Klaus Janschek: System design of mechatronic systems: methods - models - concepts . Springer, 2010, ISBN 978-3-540-78876-8 .
• Dimitri Jeltsema, Jacquelien MA Scherpen: Multidomain Modeling of Nonlinear Networks and Systems. Energy- and Power-based Perspectives . In: IEEE Control Systems . tape 29 , no. 4 , August 2009, p. 28-59 , doi : 10.1109 / MCS.2009.932927 .
• Ekbert Hering, Heinrich Steinhart: Taschenbuch der Mechatronik . Carl Hanser Verlag GmbH & Co. KG, 2005, ISBN 978-3-446-22881-8 .
• William Bolton: Building blocks of mechatronic systems - Bafög edition . 3. Edition. Pearson Studies, 2005, ISBN 3-8273-7098-1 .
• Christoph Strunk: Modern Thermodynamics: From Simple Systems to Nanostructures . De Gruyter Oldenbourg, 2015, ISBN 978-3-11-037105-5 .

Web links

• Gottfried Falk: The conceptual structure of physics. (PDF) Retrieved July 3, 2015 . 
• Friedrich Herrmann: The Karlsruhe physics course. Retrieved July 3, 2015 . 
• Werner Maurer: Wiki system physics. Werner Maurer, accessed July 3, 2015 . 
• Jörg Grabow: Overview of the constitutive laws. (PDF) Jörg Grabow, February 1, 2015, accessed on July 3, 2015 . 

Individual evidence

1. ↑ Rüdiger G.Ballas, Günther Pfeifer and Roland Werthschützky: Electromechanical systems in microtechnology and mechatronics: Dynamic design - basics and applications . 2nd Edition. Springer, 2009, ISBN 978-3-540-89317-2 , pp. 11 .
2. ↑ Jörg Grabow: Generalized networks in mechatronics . 1st edition. Oldenbourg Wissenschaftsverlag, 2013, ISBN 978-3-486-71261-2 , p. I .
3. ^ Robert H. Bishop: Mechatronics: An Introduction . CRC Press, 2005, ISBN 1-4200-3724-2 , pp. 8.4 .
4. ↑ Malcom C Smith: Synthesis of mechanical networks: the inerter . Ed .: IEEE Transactions on Automatic Control. tape 47 , no. 10 , October 2002, p. 1648-1662 .
5. ↑ Oliver Heaviside: Note 4. Magnetic resistance etc. In: Electrical papers . tape 2 . Macmillan and co., London 1894, p. 166 ( archive.org ).
6. ^ Frederick V Hunt: Electroacoustics: the Analysis of Transduction, and its Historical Background . Ed .: Harvard University Press. OCLC 2042530, 1954.
7. ↑ Heinrich Barkhausen: The problem of the generation of vibrations with special consideration of fast electrical vibrations . August Pries Leipzig, Göttingen 1907.
8. ^ Vannevar Bush's Differential Analyzer. Retrieved July 5, 2015 . 
9. ^ Walter Hähnle: Scientific publications from the Siemens group. XI. Volume first issue . Completed on March 12, 1932 (=  scientific publications from the Siemens Group . No. 1.11 ). Springer-Verlag Berlin Heidelberg, 1932, p. 1-23 , doi : 10.1007 / 978-3-642-99668-9 .
10. ^ Floyd A Firestone: A new analogy between mechanical and electrical systems . In: Journal of the Acoustical Society of America . No. 4 , p. 249-267 (1932-1933).
11. ↑ Busch-Vishniac, J. Ilene: Electromechanical Sensors and Actuators . Springer Science & Business Media, 1999, ISBN 0-387-98495-X .
12. ^ Henry M. Paynter: Analysis and Design of Engineering Systems . Ed .: MIT Press. OCLC 1670711, 1961.
13. ^ The Dresden School of Acoustics in the years up to 1990. (No longer available online.) Archived from the original on July 6, 2015 ; Retrieved July 5, 2015 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.  
14. ↑ Arno Lenk: Electromechanical Systems: Systems with concentrated parameters . 1st edition. tape 1 . VEB Verlag Technik, Berlin 1971.
15. ^ Peter E. Wellstead: Introduction to Physical System Modeling. (PDF) Control Systems Principles, accessed on July 4, 2015 .